Simulating the function of sodium/proton antiporters

Significance

Understanding the molecular nature of the function of transporters is of significant interest in view of their biological importance. However, the actual structure–function relationship of transporters is far from being clear. To advance this understanding, we simulated the activity of the Na⁺/H⁺ antiporter NhaA by a Monte Carlo approach, based on the calculated coarse-grained electrostatic landscape of the system. The simulations reproduce the observed trend and provide a powerful insight into the origin of the transporting function and the molecular basis of its control. We also show that a large conformational alternation is essential for the transporter function. The approach used should be useful for general studies of other transporters.

Abstract

The molecular basis of the function of transporters is a problem of significant importance, and the emerging structural information has not yet been converted to a full understanding of the corresponding function. This work explores the molecular origin of the function of the bacterial Na⁺/H⁺ antiporter NhaA by evaluating the energetics of the Na⁺ and H⁺ movement and then using the resulting landscape in Monte Carlo simulations that examine two transport models and explore which model can reproduce the relevant experimental results. The simulations reproduce the observed transport features by a relatively simple model that relates the protein structure to its transporting function. Focusing on the two key aspartic acid residues of NhaA, D163 and D164, shows that the fully charged state acts as an Na⁺ trap and that the fully protonated one poses an energetic barrier that blocks the transport of Na⁺. By alternating between the former and latter states, mediated by the partially protonated protein, protons, and Na⁺ can be exchanged across the membrane at 2:1 stoichiometry. Our study provides a numerical validation of the need of large conformational changes for effective transport. Furthermore, we also yield a reasonable explanation for the observation that some mammalian transporters have 1:1 stoichiometry. The present coarse-grained model can provide a general way for exploring the function of transporters on a molecular level.

Active transport lies at the heart of a plethora of important functions such as muscle contraction and neuronal activity. It is the hallmark of bioenergetics in cells and is both ancient and universal. Secondary active transporters are a subgroup of active transporters in which two different substrates are transported by the same system, with the electrochemical potential of one substrate, moving down its gradient, driving the active pumping of the second substrate. Most secondary active transporters consume the gradient of either protons or sodium ions.

Na⁺/H⁺ antiporters exchanging protons for sodium ions, are of great interest as prototypic secondary active transporters, serving as arguably the simplest case of secondary active transport. Na⁺/H⁺ antiporting is present in all domains of life and is essential in many diverse processes ranging from basic survival in bacteria (1) through salt tolerance in plants (2) to complex roles in hypertension and ischemia in mammals (3, 4).

The first high-resolution X-ray structure determined for any Na⁺/H⁺ antiporter was the structure of NhaA from Escherichia coli (5), and to date only bacterial structures have been solved (6, 7). The role of NhaA in bacteria is the removal of excess Na⁺, which could have detrimental effects on the bacterium, from the interior of the cell while maintaining H⁺ homeostasis. Not surprisingly, NhaA’s activity is highly pH dependent and is diminished at both high and low pH values (8). The presence of a “pH sensor” has been postulated and sought but with no decisive results. Recent evidence supports an alternative hypothesis: That there is no pH sensor per se and that pH dependence is caused by competition between substrate ions and protons (9, 10).

NhaA has been extensively studied biochemically, revealing that both the residues D163 and D164 are essential for activity (11), most likely serving as the binding site for both protons and Na⁺ ions. Additionally, the stoichiometry of NhaA has been determined to be 2:1, so two protons are consumed in the pumping of one sodium ion against its electrochemical gradient (12). The transporter also has been shown to function in reverse, allowing a sodium gradient to pump protons actively, if needed (13).

The basic mode of action of NhaA, as for most other active transporters, is assumed to be alternating access (14). (General considerations that support the alternating access model in the case of LacY are discussed in ref. 15.) However, despite the very reasonable assumption that an alternating access model with large conformational changes is essential for the function of transporters, we are not aware of kinetic or landscape indications that prove other gating mechanisms are impossible (SI Text). In the case of NhaA, although there is no definite evidence to prove the assumption of a large conformational change, the presence of pseudosymmetry in the structure (5, 7), which is characteristic of transporters (16), and the revelation of the structure of the Na⁺/H⁺ antiporter from Thermus thermophilus (NapA), which is considered to be in the alternative conformation (6), seem to support the model of alternating access mediated by large conformational changes. Furthermore, recent structural analyses of the related Na⁺/H⁺ antiporter from Methanocaldococcus jannaschii (MjNhaP1) (17) found two conformers that probably are the two alternating configurations. Despite this reasonable picture, it is useful to approach the transport problem without assuming a priori that the alternating conformation model is essential. Therefore, we initially explored the possibility that, as in some chloride–proton antiporters in the chloride channel family (18), NhaA might be a transporter that does not adhere to the canonical paradigm of alternating access via large conformational changes.

Computational studies that were conducted after the resolution of the structure of NhaA examined the architecture (5), selectivity (19), mechanism of action (20), and homologs of E. coli NhaA (21). Molecular dynamics (MD) simulations have been conducted on NhaA, and a mechanism of action suggesting roles for residues D163 and D164 has been proposed, (20). A recent MD study (7) tried to explore the role of ion pairing in the action of NhaA. Furthermore, a recent extensive electrophysiological study shed new light on the pH response of the protein, supported by a minimal kinetic model (22).

Although the above insightful studies have provided important advances in the understanding of the mechanism of NhaA, we still lack structure-based free-energy surfaces that are crucial for understanding the key role of the proton transport (PT) and the modulation of the pK_a of aspartic acid residues by the protein electrostatics (which include water penetration and reorganization of the protein residues). Thus, it is essential to relate the overall transporter structure to the transporter function through nonphenomenological analysis. Of course, MD atomistic simulations may lead to advances in this direction, but such approaches cannot yet provide a complete picture of the Na⁺ transport process with its coupling to large conformational changes and PT. Thus, in this work we try to advance the molecular understanding of transporters by using a strategy that allows us to investigate the energy–structure relationship of NhaA, with the aim of gaining a realistic description of the transport cycle as well as the pH response. This approach reflects our belief that the challenge of understanding the directional transport should be addressed by simulations that generate the actual relevant structure–transport relationship. Doing so requires the ability to explore the long-time transport process as a function of the pH and [Na⁺] gradients. In the present work we address this challenge, focusing on the NhaA system as a case study.

The strategy adopted here combines several approaches. We started by exploring semi-macroscopically the energetics of Na⁺ and proton binding to D163 and D164 of NhaA and obtain distinct Na⁺-binding profiles that are pH dependent. We then used the resulting energetics in Monte Carlo (MC) simulations that examine the performance of two transport models. We found that we can reproduce the observed transport features and in particular the 2:1 stoichiometry.

Results and Discussion

Charting the Landscape.

To explore the function of Na⁺/H⁺ antiporters, we consider the NhaA system (presented in Fig. 1). The basic simulation system was constructed by starting with the X-ray structure of NhaA (believed to be in the inward-open state) and embedding it in a simplified membrane model (23). The protein plus membrane system was then solvated in a water sphere by the standard MOLARIS protocol (24), minimized (with small time steps), and run in short relaxation runs with a small restraint on the X-ray structure using MOLARIS. Several such locally relaxed structures were used for all subsequent calculations and were found to give stable results.

Fig. 1.

Presentation of the simulation system. The protein appears in orange, and the membrane particle grid is shown as cyan spheres. The membrane has been truncated so it does not obscure the protein. Residues D133, D163, D164, and K300 are shown in a licorice presentation.

The first step of our study involved calculations of the pK_as of key acidic residues in NhaA, using the semi-macroscopic protein-dipole–Langevin-dipole linear response approximation (PDLD/S-LRA) protocol (SI Text and ref. 25). The pK_a values of D163 and D164 were computed, in each case with the other residue being either protonated or charged (without the transported Na⁺). The initial calculations were performed while the rest of the protein ionizable groups were kept at their neutral state, given the effect of distant residues macroscopically with a large dielectric constant (ε_p) (25). The results obtained are summarized in Table S1. The pK_a values for D163 and D164 were fairly similar, at ∼7. Next, we focused on the roles of the adjacent residues D133 and K300, which have been suggested to be important for NhaA function (5, 11), and computed pK_a values of −2.15 and 12.81, respectively, indicating that both are definitely found in their charged form in the protein. Taking this result into account, we repeated the pK_a calculations of D163 and D164 with D133 and K300 explicitly charged; the corresponding results also are summarized in Table S1. Encouragingly, the results of the second computation provided a significant improvement in the agreement with the experimentally observed pH dependence (as described below). Therefore we used the pK_a values of 9.3 for the first protonation and 7 for the second; these values reflect the average of the calculated results obtained for the two different aspartic acid side chains (Table S1). The pK_a relation to the Na⁺ binding is discussed further in SI Text.

Table S1.

pK_a values for D163 and D164

System	First protonation		Second protonation
	pKa of D163 (D164−)	pKa of D164 (D163−)	pKa of D163 (D164 protonated)	pKa of D164 (D163 protonated)
Implicit charges	7.28	7.89	4.66	6.81
D133, K300 explicit	8.4	10.3	6.5	7.5
D133L (K300 explicit)	5.66	5.22	4.29	5.8
D133N (K300 explicit)	5.98	7.55	2.52	5.65
K300M (D133 explicit)	15.29	11.2	10.27	9.39

The protonation state of the other aspartic acid residue is shown in parentheses. pK_a values include the membrane region correction as explained in SI Methods and Systems.

Reliable pK_a calculations are quite crucial for modeling the function of transporters, and it is very hard to accomplish such a task with empirical approaches such as the method applied in the popular PROPKA server (26) used by Lee et al. (7). This method is not based on the actual electrostatic free energy in protein interiors, nor has it been validated with relevant experimental results in protein interiors (the validation issue is discussed in ref. 27). The issue is of equal importance when one considers the proposal of ion pairing (e.g., with K300), which is based on PROPKA-type calculations. Exploring such a crucial issue requires proper modeling of the protein compensating dielectric, which includes water penetration and reorientation of polar groups and which cannot be obtained without proper sampling and LRA or extensive free-energy perturbation studies (e.g., see the discussion in ref. 27). Of course, pK_a calculations should be judged by their ability to reproduce the observed pH profile, as presented below. One may try to use Poisson–Boltzmann (PB) calculations, but only if such calculations reflect the proper ε_p and include an LRA or equivalent treatment (SI Text).

Next we examined the binding of substrates by using the PDLD/S-LRA method to calculate the energetics of Na⁺ binding by evaluating the solvation energy of bringing an Na⁺ ion from the bulk to several points within the protein’s cavities along a vector from the cytoplasmic or periplasmic bulk regions to the binding site. The binding site was chosen to be a point in the vicinity of D163 and D164. Although this point does not necessarily represent the exact physiological binding location (which has not yet been determined by structural means), it is likely to be very close, and the MD relaxation is expected to lead the ion into a position that represents the actual binding position (also see SI Text).

The energy curves were calculated for four different protonation states: both D163 and D164 in the protonated form (HH); only one of them in the protonated form [(H−) and (−H), henceforth marked as (H−) for simplicity]; and both D163 and D164 in the charged form (−)(−). The results, shown in Fig. 2, express three discrete types of energy curve. One is the Na⁺-repulsion curve, seen for HH, in which the energies rise steeply as the Na⁺ ion is brought into the protein cavities. Such an energy profile creates a very high barrier for the Na⁺ ion, and this barrier plays a major role in helping understand the transport cycle (see below). This result is a logical one, because when no residue is charged, there is no electrostatic attraction between an Na⁺ ion and the binding site. Another type of binding curve is the electrostatic trap, seen in (−)(−), with moderate barriers but a deep trough, binding strongly and potentially trapping the ion. This result also is reasonable, because the charge of two adjacent charged aspartic acid residues would apply a strong electrostatic force on an Na⁺ ion. The third curve is for intermediates, seen for both H(−) states, which show similar but less extreme cases of the HH and (−)(−) curves. An important characteristic of all the curves, as discussed in SI Text, is the relative symmetry with respect to the z axis. It is encouraging that the calculated K_d for the values obtained for the fully charged state is in the range of the experimentally measured apparent K_M of NhaA (Table S2).

Fig. 2.

The PDLD/S-LRA profiles for Na⁺ binding, for the different protonation states of D163 and D164 in NhaA with error bars (n = 10; ε_p = 4).

Table S2.

ΔG_bind for Na⁺ and the corresponding calculated K_ds

Protonation state	ΔGbind, kcal/mol	Kd, mM
	Calculated (PDLD/S-LRA)	Calculated (from PDLD/S-LRA)
HH	11	—
H(−)	7	—
(−)H	−2	35.7
(−)(−)	−5	0.24
	Calculated (from observed KM)	Observed
Literature (22)	−3.43	3.3*
Literature (23)	−3.79	1.8*

^*Apparent K_M.

Next, we looked for an effective strategy for using the ion transport landscape in performing structure-based analysis of the transporter function. As a first step, we tried to generate a state diagram and use it to explore different mechanistic pathways for the action of NhaA. This strategy, which has been very useful in studying PT (28), resulted in a chart that contains the most important combinations of the binding or unbinding of Na⁺ or protons from either side of the membrane (Fig. S1). In the present case, however, the use of such a chart to determine the resulting mechanism is too challenging because of the difficulty of assessing the probability for each path. Thus, we preferred to continue the analysis with an MC treatment that seemed to provide more insight in exploring the molecular nature of the transition over the landscape of the system (SI Text).

Fig. S1.

State diagram describing the possible pathways. NhaA is presented as a blue square, and the membrane is represented by a yellow rectangle. Particles and binding sites are presented as circles. Highlighted membrane rectangles represent the conceptual beginning and ending of a cycle. Black arrows mark the desired cycle, and green arrows depict cycles that may lead to a 1:1 stoichiometry. Red arrows depict cycles that lead to proton or Na⁺ leaks, and brown dashed arrows depict steps that are hidden for visual clarity. The numbers within the membrane rectangle represent the relative free energy (taken from the MC energies and rounded to an integer), and the numbers along the arrows represent height of the energy barrier (in kcal/mol). Gradients of 2 kcal/mol are used.

Two MC models were considered (Fig. 3). In the first, model 1, the transporter was kept with identical conformational openings on both sides, meaning that the barriers on both sides of the transporter were as seen in the PDLD/S-LRA calculations for NhaA (in which the effective profile is based only on the protonation state). This model, which corresponds to the scheme in Fig. 3, Upper, is based on the assumption that the protein has a single conformation, similar to the X-ray and the relaxed structures. Of course, the LRA calculation in the PDLD/S-LRA method allows the protein to undergo small changes, such as reorientation of dipoles, which could be enough to account for differential energy barriers on either side of the membrane, allowing effective different conformations as classically expected from a transporter. However, on the larger scale the structure assumes the same architecture. The actual MC simulations used a simplified surface that is outlined in Fig. S2.

Fig. 3.

Schematic representations of the models explored. The transporter is depicted in blue, and the membrane is depicted as a yellow rectangle. D163 and D163 are depicted as green circles; Na⁺ ions are depicted as yellow circles. The arrows indicate the particle motion. Each scheme represents a step in a sequential transport cycle, according to the model, from left to right. Model 2 also illustrates the inverted bananas principle graphically.

Fig. S2.

Energy profiles. (A) A presentation of the energies used in the MC simulations. Binding sites and barriers are marked by circles. (B–D) The alignment between the energies obtained with the PDLD/S-LRA analysis (Fig. 2) and the energies used in the MC simulations for the HH (B), H(−) (C), and (−)(−) (D) protonation states.

As shown below, model 1 could not reproduce the transporter function under a pH gradient, so we explored model 2, in which the transporter undergoes larger conformational changes that could not be detected by the relatively short simulations of the PDLD calculations or even by longer free-energy perturbation calculations. This model assumes, and in some respects tries to verify (SI Text and Fig. S3), that the transporter has alternating access configurations in the form of “inverted bananas” (Fig. 3, Lower; also see the “rocking bundle” in refs. 6 and 29), resulting in the opening of one side concomitant to the closing of the opposite side. (For more elaborate considerations of conformational changes of transporters, see refs. 30 and 31.) It is assumed that the conformations are separated by a high-energy barrier (see below), considering three conceptual conformations: inward-facing, outward-facing, and occluded, as observed previously for other transporters (e.g., ref. 32). The alternating structures were built into the effective landscape (in terms of the corresponding energy profiles), and the height of the conformational barrier was estimated by assuming that the corresponding step is the rate-determining step and that the corresponding rate constant is similar to the observed rate of ion transfer of about 1–1,500/s [the reported experimental exchange rate of NhaA is between ∼100 and 100,000 ions/min (8, 12)]. When the time of downhill trajectories is considered (SI Text), this model led to a barrier of ∼14–16 kcal/mol for the conformational change. This barrier was incorporated into our time-dependent MC scheme, increasing the barrier by 2 kcal/mol so that we can use the same MC frequency of ion movements (SI Text). To save computational time and considering that no binding or unbinding occurs on the occluded state, we omitted this state from the MC moves. Furthermore, because barriers in the range of 16–18 kcal/mol require extremely long simulation times, and because the results converge with a barrier beyond a threshold of ∼8 kcal/mol (Fig. S4), the simulations were performed using barriers of 8–10 kcal/mol for the conformational change (SI Text).

Fig. S3.

OC analysis. The energy of hydration for an Na⁺ ion being charged from 0 to +2 in the HH protonation state. Similar results were obtained with the other protonation states but are omitted for visual clarity.

Fig. S4.

Convergence of calculated stoichiometry with the barrier of the conformational change. Ratio of Na⁺ and H⁺ flux in the MC simulations as a function of the energy barrier used for the conformational change (Results and Discussion, Charting the Landscape). The Inset shows an enlargement of the more relevant values. Error bars represent SEM (n = 25).

Reproducing the pH Dependence for Cases with No pH Gradient.

The first step in the detailed analysis of our models was the analysis of the pH dependence of the Na⁺ transport in the absence of a pH gradient. Here we tried to reproduce the recent results of Mager et al. (13), who measured the Na⁺ flux in NhaA as a function of pH and found a dramatic decline at both high and low pH.

Our study used the MC model with energy values that correspond to the PDLD/S-LRA results given above (Fig. S2). The calculations involved MC simulations with equal pH on both sides of the membrane and with an Na⁺ gradient. Looking at the Na⁺ flux in the MC simulations (also see SI Text), our results were in reasonable agreement with Mager et al. (Fig. 4), with a peak at pH ∼6 and a bell width very similar to the one presented by Mager. These results correspond to Mager’s observations, with a modest shift in the optimal pH; the overlap of the bell curves is in excellent agreement, suggesting that the model reproduces the results well but that the pK_as used are not perfect. Although this finding is encouraging, one can ask whether the same trend can control the transporting function as well, in terms of Na⁺ flux.

Fig. 4.

The pH response in the MC simulations. Normalized net flux of Na⁺ in the MC simulations (red trace) compared with the experimental results of Mager et al. (13) (blue trace). Insets show the energy profiles of the MC (using the same colors as in Fig. 2) as a function of the pH in that point on the chart. For clarity, only some of the states are depicted by a schematic square representing NhaA. The profiles are translated vertically to account for the differences in energies caused by the protonation state, based on the averaged pK_a values and the corresponding pH. The results are shown for model 2, but similar results were obtained with model 1. Note that the x axis has different values for each curve.

We note that although our calculated pK_a values result in a shift to an acidic optimal pH, compared with the experimental results, when we calculated pK_a values using a microscopic adiabatic charging approach (with a polarizable force field), we obtained pK_as of around 10–12. Although this fully microscopic approach tends to overestimate the pK_a of internal groups (33), the results indicate that the actual pK_a values should be somewhat higher than the ones we used in the MC models, resulting in better agreement with the pH profile.

The reason for the success of our MC models in the absence of a pH gradient is instructive. As is clear from Fig. 4, in the low pH range the HH state is lower in energy, and the Na⁺ transport is blocked, whereas at high pH the (−)(−) state has the lowest energy, and the Na⁺ is trapped. Thus, only the intermediate states have an optimal flux

Reproducing and Understanding the Transport Function.

After obtaining the observed trend for the case without a pH gradient with both models 1 and 2, we moved to the more challenging test, examining which model could account for the antiport function, i.e., the transport of Na⁺ in the opposite direction of a proton gradient. MC simulations using model 1 were conducted with a proton gradient but no Na⁺ gradient. No correlation was found between the proton and the Na⁺ fluxes (Fig. 5). Careful analysis of the system revealed the likely cause for this problem: With a transporter that is open to both sides of the membrane simultaneously, the protonation state of the transporter is dictated by the lower pH of the two adjacent spaces, so the higher pH on the other side has little to no effect. This conclusion is very important because it indicates that the alternating access model is essential. (Although the current paradigm for transporters incorporates the alternating access model, we thought it was instructive to consider a model that did not include this feature to determine the requirement for alternating access without initially assuming it.)

Fig. 5.

The results of the MC simulations of the two models, showing the net calculated flux of the Na⁺ and H⁺ ions. 0 represents no pH gradient; + and − represent a pH gradient created by a difference of 2 pH units in opposite directions. Each arrow represents the average of 25 simulations, proportionally scaled down for visual clarity. The dashed line emphasizes the experimental stoichiometry.

The MC simulations using model 2, with a proton gradient and no Na⁺ gradient, resulted in a much lower proton flux than seen with model 1 and in a biased flux of Na⁺ in the opposite direction (Fig. 5). The lower flux for protons reflects the closed position of the transporter in some of the MC steps, resulting in less proton movement and thus in less flux. Repeating the calculations with a pH gradient in the opposite direction or with an Na⁺ gradient and no proton gradient produced the fluxes shown in Fig. 5 and Fig. S5, respectively. The active Na⁺ flux is a result of the energetics as described by our PDLD/S-LRA calculations. In brief, when the transporter is exposed to the more acidic side, the HH state becomes predominant and promotes Na⁺ unbinding; when exposed to the other side, it assumes the (−)(−) state and promotes Na⁺ binding. These states effectively gives NhaA different affinities for Na⁺ on both sides of the membrane, an efficient scheme for transport (34). Over sufficiently long simulations, a net antiport outcome with a stoichiometry of ∼2:1 is expected. These conditions are expected only under a fairly narrow range of pH values, within which, indeed, NhaA functions (albeit with a pH shift; see above). However, the stoichiometry that was obtained was higher than 2:1 (Fig. 5), probably because the global minimum on each side is not occupied 100% of the time (Movie S1). However, in physiological conditions some extent of substrate leakage is probably inevitable (34).

Fig. S5.

Na⁺-driven proton pumping. MC net fluxes for Na⁺ and protons with an Na⁺ gradient of 2 kcal/mol and no proton gradient at different pH values. Negative values denote flux in the opposite direction. Error bars represent SEM (n = 25).

To clarify the origin of the above results, we plotted our free-energy landscape, considering proton and Na⁺ binding as the two reaction coordinates (Fig. 6). The surfaces reflect the alternating access model by having a high barrier on only one of the sides at any given time. Examination of the surfaces immediately reveals the mechanism explained above. When the transporter is open to the acidic side, two protons will bind, and an Na⁺ ion (if present) will unbind. When the transporter is open to the basic side, the two protons will unbind, and an Na⁺ ion will bind instead. The cycle can be summarized as 2H⁺:1Na⁺ antiporting. In our model a deviation from the global minima would result in a deviation from the perfect 2:1 stoichiometry, but we do not expect this deviation to have a substantial effect when averaged over large numbers of cycles.

Fig. 6.

The effective landscapes for the transporting function. (A) A 3D surface of model 2. For simplicity the proton axis depicts two sequential proton bindings (see Results and Discussion, Reproducing and Understanding the Transport Function), as shown by the squares representing NhaA. The left and right surfaces represent the inward-facing and outward-facing states, respectively. (B) The same surfaces shown in 2D color. For visual clarity energies above 10 kcal/mol were truncated to black color. Squares representing NhaA are shown to represent only some of the states. Dashed circles highlight the global minima. The arrows depict the path of a perfect transport cycle.

Our study provides an interesting hint regarding the stoichiometry of mammalian Na⁺/H⁺ exchangers (for a review, see ref. 35), which, in contrast to bacterial NhaA, is 1:1. We note that the extremely conserved DD motif in NhaA (D163 and D164) appears as ND in many Na⁺/H⁺ exchanger (NHE) members (36). Thus, following our model it is reasonable to expect that the stoichiometry of 2:1 seen when two aspartic acid residues participate may translate to 1:1 when only one aspartic acid residue is present. This notion has been supported by our MC simulations in which we removed one of the aspartic acid residues but kept the other binding curves of NhaA unchanged (Fig. S6). However, a full verification requires the still unavailable high-resolution structure for NHE and the corresponding calculated binding profiles, as well as accounting for the different pH-activity profile of NHE-type transporters. In this case we would need a binding site that provides a strong binding for Na⁺ without having a second ionized acid, as well as a pK_a for a single aspartic acid that would allow exchange in the observed pH profile of NHE. Interestingly, the recently reported structure of the MjNhaP1 transporter (17) offers an opportunity to explore this issue, because this protein probably also supports a 1:1 stoichiometry and has a binding site with ND instead of DD. This point is considered in SI Text and should be explored further in the future.

Fig. S6.

Antiporting stoichiometry for hypothetical mammalian NHE. MC net fluxes for Na⁺ and protons with a pH gradient. The simulations were performed using model 2 with conceptual energetics of a single aspartic acid residue (see Reproducing and Understanding the Transport Function in the main text). Each arrow represents the average of 25 simulations.

When we consider the sequence of binding of either the aspartic acid residue or the Na⁺ ion, our pK_a values suggest that D163 is the first residue to lose a proton or the last to bind one (Table S1) and that an Na⁺ ion would be expected to bind only after D164 becomes deprotonated (Fig. 2). Therefore, the energetically logical sequence of events starting from the HH state would be deprotonation of D163 and then an exchange between the proton of D164 and an Na⁺ ion, a sequence very similar to that suggested by Arkin et al. (20).

The pathway of a perfect transport cycle is presented in Fig. 6B. We note that this model is based essentially on direct competition between the Na⁺ ions and the protons, where protonation of D163 and D164 results in Na⁺ repulsion, and deprotonation results in Na⁺ attraction. Competition between substrates has been suggested and shown biochemically for NhaA (9, 37).

Overall Considerations of the Mechanism of Action.

The present simulation study appears to provide a working model for the transport cycle of NhaA and perhaps other transporters that function in a similar way. With the presence of a pH gradient across the membrane, the transporter alternates and is exposed to a different pH at different times. When facing the more acidic side, the transporter undergoes protonation of both its aspartic acid residues, resulting in the unbinding of the Na⁺ ion (or no binding if no Na⁺ was bound initially). Because this is the lowest energy state, it is expected to be maintained (on average) until the transporter undergoes a conformational change to face the more basic side of the membrane. Then, the transporter will release the two protons, a process that could be enhanced by Na⁺ binding. Energetically, the most stable state is fully deprotonated with a bound Na⁺ ion. Being stable, this state is maintained (on average) until another conformational change ensues, repeating the cycle at a 2:1 stoichiometry. Both binding of Na⁺ to the double-protonation state and releasing of Na⁺ from the double-charged state would lead to leak and loss of energy. These states, however, are unfavorable. The overall cycle is presented in Fig. 7.

Fig. 7.

A schematic presentation of the working model suggested in this work. NhaA is depicted as two banana-shaped domains, alternating access through an inversion process. The circles represent particles and binding sites.

Concluding Remarks

The present simulations using structure-based models reproduced the observed transport features of the NhaA system. It was found that the fully charged protein acts as an Na⁺ trap, whereas the fully protonated protein poses an energetic barrier too high for the ion to cross. It is through the alternation between these states, mediated by the partially protonated protein, that Na⁺ and protons can be exchanged across a pH gradient at 2:1 stoichiometry. Interestingly, our study provides very strong support for the reasonable but unproven alternating access model for NhaA.

A key feature that has been strongly supported by the current work is the requirement of conformational changes. This effect appears to be essential for reproducing the observed transport properties. Furthermore, we established the origin of the difficulties with models that do not include conformational changes. Obviously it would be important to explore the details of the conformational change, as was attempted by Lee et al. (6), and to produce a detailed structure-based energy landscape. In this respect our progress has been somewhat limited. We found strong support for the conformational model by the overcharging calculations (SI Text and Fig. S3). However, a more direct landscape calculation would require the generation of the path between the two conformations and exploring the Na⁺ and H⁺ profile along this path. Interestingly, the equivalent of the overcharging effect may play some role in coupling the conformational change to the flow of Na⁺ and H⁺, but this effect is not expected to be a major one as long as the conformational barrier without the coupling is sufficiently high.

The stoichiometry obtained in our study deviates from 2:1 as a function of pH and is pK_a dependent. This result actually might hold true for NhaA and has not been tested thoroughly enough. That is, the original work on NhaA (12) measured ratios of nearly 2:1 within a fairly narrow range of pH values; the pH range was restricted for technical reasons in play at the time of that investigation, and to the best of our knowledge no work has tested stoichiometry at pH values beyond that range. Therefore, further work testing varying pH values on both sides of the membrane is required to validate or disprove this prediction.

Our model attributes a passive role to the Na⁺ ions. That is, although an Na⁺ gradient alone creates a biased inverted proton flux (SI Text and Fig. S5), PT still takes place in the absence of Na⁺. This trend seems to be in contrast to the corresponding experimental behavior of NhaA. One optional explanation is that the conformational changes discussed above are coupled to the Na⁺ and its solvation, and protons by themselves (H₃O⁺) cannot perform this action efficiently. For example, it is possible that the conformational barrier is different when the protein is found in the (−)(−) configuration or in the (HH) configuration. This option has been supported by our MC simulations (Fig. S7), but the interesting issue of electrostatic coupling has not yet been explored by structural studies.

Fig. S7.

Na⁺ coupling. MC net fluxes for protons with a pH gradient in the presence of absence of Na⁺ ions. These simulations were performed using model 2 with the exception that the barrier for alternating conformations was increased by 6 kcal/mol for the HH and H(−) states without an Na⁺ ion (see Concluding Remarks in the main text). Error bars represent SEM (n = 25).

In relating our simulation studies to other current strategies, one should realize that using powerful MD simulations to explore ion penetration and ion-pairing effects (e.g., ref. 7) does not necessarily provide information about the action of complex systems such as transporters. The analysis of functions must be based on calculating free-energy surfaces and free-energy profiles, which are very hard to obtain even with simulation runs in the microsecond range. Without a free-energy surface, we cannot explore the proposal in ref. 7 and thus cannot confirm or disprove it.

The progress presented in this work in achieving a detailed understanding of the control of the NhaA transport function provides more than just an intellectual insight. The Na⁺/H⁺ antiporter itself is of high importance in medicine. Bacterial Na/H antiporters potentially could be used as a drug target in cases in which NhaA is essential for the pathogenicity of bacteria (38). Perhaps more importantly, the human equivalent of NhaA, NHE, is associated with a growing number of human pathologies, such as cancer, hypertension, and reperfusion damage, among others (for a review, see ref. 3), so the study of NhaA has potential pharmaceutical significance.

Methods and Systems

The NhaA simulation system was constructed by taking the protein from the Protein Data Bank (PDB ID code: 4AU5) (7). A single monomer was aligned so that the z axis corresponds roughly to the membrane normal, and all atoms and molecules except for the protein itself were removed from the system. A 3D grid of nonpolar particles, using a spacing of 3 Å, was added around the protein, representing a simplified membrane with dimensions of 66 × 66 × 30 Å. The system was first minimized using MD with small time steps and low temperature and was then relaxed with several 20-ps runs subjected to small restraint on the X-ray structure. The several resulting structures were used as starting points for the subsequent PDLD/S-LRA calculations (described in SI Text). The calculations were performed with the MOLARIS program (24).

To explore the possible role of conformational changes, we adopted the overcharging (OC) idea (33), comparing NhaA and NapA. First, we constructed a simulation system for NapA (PDB ID code: 4BWZ) (6) as described above. Then, we took the same starting structures used for the binding calculations, placing an Na atom in different points along the binding path and changing the atom’s charge from of 0 to +2 using the adiabatic charging method of the MOLARIS package (SI Text and ref. 24).

A key element of the present work has been a time-dependent MC simulation similar to the one used in our previous studies of PT (for details, see SI Text) (39). The construction of the energy surfaces is detailed in SI Text.

SI Methods and Systems

This section provides additional details on the simulation methods, clarifying the actual procedures used.

The crucial pK_a values of D163 and D164 were evaluated using the semi-macroscopic PDLD approach as implemented in the MOLARIS package (24, 25) (see below), using the systems described in the main text, with a region II radius of 18 Å. We note that a sphere with this radius could not contain a uniform membrane slab (for a consistent treatment, see ref. 40). Thus, for computational convenience, we examined the effect of changing the radius from 18 to 34 Å and estimated from the calculations that the pK_as obtained from the 18-Å calculations should be corrected by adding 1 pH unit to each pK_a computed.

The semi-macroscopic simulations were done with the PDLD/S-LRA model; this model takes the original PDLD model (24) and scales the corresponding contributions by assuming that the protein has a dielectric constant, ε_p. This assumption is formulated in a cycle that arguably provides the clearest link between the microscopic and macroscopic models. To reduce the unknown factors in the ε_p, it is useful to move to the PDLD/S-LRA model, in which the PDLD/S energy is evaluated within the LRA with an average of the configurations generated by an MD simulation of the charged and uncharged states (41). In this way, the MD simulations can be used to generate protein configurations for the charged and uncharged forms of the given “solute” automatically; then these contributions can be averaged. Because the protein reorganization is considered explicitly, there are fewer uncertainties regarding the ε_p (27). More specifically, the Na⁺ ion was inserted at the putative binding site of the relaxed protein (at the vicinity of D163 and D164 for NhaA and at intervals of 1–2 Å along vectors leading from the binding site to the cytoplasmic or periplasmic bulk, based on visual inspection of the protein). The positions of the ions were refined, moving them in the x–y plane, to avoid van der Waals overlap with the protein atoms. A relaxation step was performed for 10 ps before the PDLD calculation, and 10 configurations were taken from this relaxation run and used in the PDLD calculation at 1-ps intervals. One may try to use PB calculations, but only if such calculations reflect the proper ε_p and include an LRA or equivalent treatment (see discussion in ref. 25).

To corroborate the importance of residues D133 and K300 in the modulation of the pK_a, we mutated the residues to van der Waals equivalent, nonpolar residues (namely Leu and Met, respectively) and computed the pK_a values for D163 and D164. As can be seen in Table S1, the pK_a values change because of the mutations, and the effect of the mutation is much more substantial than using explicit or implicit charges for D133 and K300. Additionally, the charge of D133 is tested using the D133N mutation, which also exhibits pK_a values that would not allow proper transport, using our model.

Finally, we note that the intrinsic pK_as effectively change upon the binding of Na⁺, but changing the pK_as while taking the binding energy of Na⁺ into account would result in double counting of this energy contribution. Hence, we used the intrinsic pK_as and incorporated the effect of the Na⁺ binding in the consideration of the binding free-energy profile.

Faced with a complex landscape with many competing paths, we had several simulation options. The first was a master equation that deals with the average time-dependent probability. Such approach has been used previously, for example in our studies of PT in the bacterial reaction center (42). However, dealing with average probability forfeits insight about the motion of the individual particles and can obscure the underlying physical picture. A more physically sound approach is provided by our renormalized Langevin dynamics (LD) approach (43). This approach has been used in many of our studies, starting with the simulation of the transport in the crystallographically sited activation channel (KcsA) (44) and moving to many other systems, including F₁-ATPase (45). Other approaches have been used effectively by Oster and coworkers (e.g., Liao et al., ref. 46), although they used phenomenological rather than structure-based landscapes. However, as we realized in our study of PT in cytochrome c oxidase (39), it is possible to obtain significant insight and very long time simulations by time-dependent MC. A similar idea has been used before with Gillespie kinetic MC-type simulations (47), but the main point in our approach is that we look for a clear physical basis for the time step that is assigned for the MC moves. The assignment of the time step is quite rigorous in the case of PT along water chains (48) but becomes more qualitative in the case of Na⁺ movement or conformational changes. To achieve a relatively uniform treatment, we divided the landscape into regions with a few sites separated by barriers and then tried to evaluate the effective time-dependent behavior. More specifically, if we stand at the bottom of a barrier, the first passage time (the time of moving from the point before the barrier to the point just after the barrier (x₋ and x₊, respectively) can be approximated by

$$k_{-+}^{-1}={\tau }_{fp}(x_{-}\to x_{+})=P\left(E\right){\tau }_{dh}$$

where $P\left(E\right)$ is the probability of reaching the top of the barrier, and ${\tau }_{dh}$ is the speed of a downhill trajectory evaluated by over-damped LD for the simplified system or by MD for the explicit system. Here the idea is that the time required for the rare productive trajectory is equal to the time required for a downhill trajectory. In this case we can obtain the approximated first passage time by using MC moves with the Metropolis acceptance (49), which considers the energy difference for the given move. The approach can be refined further with more sites and with LD or MC treatment for moving in the low barrier region, but the rate-determining steps still will be determined by the high barriers. In the present case, we estimated the effective ${\tau }_{dh}$ by looking at the empirical valence bond LD study (figure 6 in ref. 48, which basically is in agreement with transition state theory) for PT (getting ${\tau }_{dh}$= 0.2 ps). In considering Na⁺ transfer, we used the system of Dryga and Warshel (50) and obtained ${\tau }_{dh}$ of about 0.3 ps from the velocity autocorrelation and finally ${\tau }_{dh}$= 10 ps for typical conformational changes in proteins. This number represents a rough compromise between LD runs with typical barriers and typical frictions (having the actual conformational reaction coordinate for our system would provide a more solid estimate). Of course, in the future more careful studies should be made with a renormalized LD treatment.

In performing the MC procedure, we assigned the same time steps (0.25 ps) for Na⁺ and the proton, and because the conformational change step is about 30 times longer, we incorporated the difference in the effective activation barrier. Therefore, for an estimated rate of conformational change of about 1/s (see main text) and ${\tau }_{dh}$=10 ps, we have a barrier of about 16 kcal/mol (see the equation above). However, in the MC procedure this barrier should be increased to about 18 kcal/mol if we attempt the moves for the Na⁺ and the conformational changes at the same frequency. Another issue is the length of the MC simulation: Even with the effective MC procedure, it takes a very long time to simulate processes in the second range. Fortunately, once the barriers are sufficiently high the simulated results scale like the corresponding Boltzmann factor, and we can use lower barriers and then scale the results back to the actual barrier heights (48). We followed this procedure in the present work after verifying that we obtained the same results while changing the height of the conformational barrier.

In NhaA the flux corresponds to a movement of ∼100–100,000 ions/min (8, 12), and it is reasonable to ascribe the rate limiting of this process to the conformational changes. Thus, with the considerations detailed above, we can assign a barrier of 18 kcal/mol for the slow flow and about 16 kcal/mol for the fast flow.

The protein landscape was simplified by interpolating the energetics of the charged particles (sodium or protons) along the transporter, from the cytoplasm to the periplasm, through the binding site (Fig. S2). The relative energies of these sites were taken from the PDLD/S-LRA results, and the energies in the bulk were adjusted to accommodate differences in pH and Na⁺ concentration. Each site (except for the bulks) could be occupied by only a single particle at a given time; the barriers could be occupied only transiently. That is, a particle on a barrier was always selected to be moved in the subsequent step. Otherwise, at each step, a single particle was selected to move at random in a random direction. When the attempted movement would result in overpopulating a site, the step was ignored, and a new step was attempted. To accommodate for the statistical bias (e.g., the chance that a proton will occupy a specific site is half that of Na⁺ because there are two proton-binding options), extensive MC simulations were performed, and corrections were added to the energies until the distribution of the particles matched the Boltzmann distribution. These corrections were tested over a very large range of different energies, and all results were self-consistent when the same correction values were used. To avoid depletion, the number of particles in the two bulk regions was equalized manually throughout the simulation; the effect of this equalization was tested and produced no bias. Finally, MC simulations of 100,000,000 steps were repeated 25 times for each MC study. In each simulation, the net flux of particles from one bulk to the other was counted for Na⁺ and protons.

The overall flux was evaluated by counting forward and reverse transport events in the MC simulations, and reporting their subtraction product. Because this value depends on both the length of the simulation and the height of the conformation change barrier, we reported the results while proportionally reducing the values obtained for visual clarity, so that all experimental values are within the same order of magnitude and can be compared (except for the pH profiles, which in all experiments were normalized by the same value, so that the highest value is 1).

The energy surfaces for the MC simulations were constructed using Gaussian functions that smoothly reconstructed the PDLD/S-LRA. Each site, as seen in Fig. S2A, is represented by a Gaussian of the same height and a uniform width (representing one so-called MC site). The function was made with five dimensions: the positions of the Na⁺ and two protons, the conformation, and the corresponding total system energy. To visualize the function, as seen in Fig. 6, the two protons were combined in one coordinate (representing sequentially two proton-binding events from the same side and then the release of the two protons to the other side), and separate surfaces were drawn only for complete outward-facing and complete inward-facing conformations. The conformation energy was manifested by barriers on the closed side. Finally, the pH gradient was implemented by modifying the energy of protons in the bulk (using pH = 7 as the reference for 0 kcal/mol).

SI Results and Discussion

Trying to Support the Conformational Transition Landscape.

As stated in the main text, the alternating access model is logically assumed to be essential for the action of transporters. However, one can wonder whether very large conformational changes are absolutely essential for effective transport. That is, although some accept the alternating access model as a given fact, others, including leaders in the field (15), still provide detailed arguments in attempts to support this model. Furthermore, in considering the origin of the gating for the Na⁺ ion, one may assume that proper synchronization between the protonation of the ionizable groups and the Na⁺ transport can create a special gating mechanism. This issue is interesting, because in cytochrome c oxidase, for example, there are no clearly observed large conformational changes, and many studies (e.g., ref. 28) have tried to generate models in which the gating is controlled by a proper coupling between protons and electron transfer. In our view, given the complexity of attempting to prove analytically the negative concept that a transporter can never work without large conformational changes, it is useful to show that in specific cases, such as the one presented here, the conformational changes are essential.

Of course, in addition to our MC demonstration that the model with large conformational changes (model 2; Results and Discussion, Charting the Landscape) is needed, the argument would be more convincing if there were direct structural support or support from the calculated landscape. However, structural information about the two alternative conformations is lacking. One could attempt to generate the two conformations by extremely long simulations, but such simulations are beyond the scope of the present study. Alternatively, we tried to compare the behavior of NhaA and the protein NapA, which is assumed to be in the alternative conformation (6). Trying to extract information from the two available structures, we performed an OC study changing Na⁰ to Na⁺² along the binding path for both molecules. The corresponding calculated results (presented in Fig. S3) demonstrate that the hydration energy for charging a particle along the cavities of NhaA shows a stark asymmetry, with the cytoplasmic cavity being much more accessible to water than the periplasmic one. The same analysis performed for NapA shows a mirror image of this behavior, in which water is much more accessible to the periplasmic cavity. This finding is in line with the suggestion that NhaA was crystallized in its inward-facing conformation and that NapA was crystallized in its outward-facing conformation. We note that although the sequence homology between NhaA and NapA is relatively low, they share highly conserved residues that are important for function and have similar architectures (6). The use of the OC approach allows us to enhance the asymmetry in charge-induced water penetration without conducting extremely long simulations. It also is likely that the OC-induced water-penetration effect mimics the actual conformational changes.

Some Functional Considerations.

In most cases, NhaA functions physiologically as a proton-driven Na⁺ pump, but it is known to be able to pump protons using an Na⁺ gradient as well. In fact, most experiment designs using it exploit this ability. To assess whether our model can accommodate this ability as well, we looked at the relation between the Na⁺ flux and the proton flux, when pH was equal on both sides of the membrane. As seen in Fig. S5, when model 2 is used, a significant inverse correlation exists between the two fluxes, at a broad pH range. The mechanism behind this inverse correlation is very similar to the one discussed in the main text. When the transporter is exposed to the high [Na⁺] side, the concentration of Na⁺ promotes the binding of sodium, and consequently protons are released [because (−)(−) is the lowest state in energy when Na⁺ is bound]; when the transporter is exposed to the low [Na⁺] side, the Na⁺ is released, and protons can bind again. However, we note that the stoichiometry in these cases is close to 1:1.

Although our model reproduces the experimental results nicely and does not allow the transport of Na⁺ ions at pH below 4 or above 10, the MC simulations allow efficient transport of Na⁺ at a pH gradient with bulk values of 3 on one side and 11 on the other, or a similar range. Transport at these pH values probably does not occur in vitro (to the best of our knowledge these conditions have never been tested and probably do not exist in vivo); nevertheless, we believe that our model gives meaningful and consistent information about the mechanism of action of NhaA.